RAITM. Chatterjee1 Public Key Algorithms ……... RAITM. Chatterjee2 Public Key Cryptography Two keys Private key known only to individual Public key available.

RAIT M. Chatterjee 1

Public Key Algorithms……..


Public Key Cryptography Two keys

Private key known only to individual Public key available to anyone

Idea Confidentiality:

encipher using public key, decipher using private key

Integrity/authentication: encipher using private key, decipher using public one


Requirements

1. Given the appropriate key, it must be computationally easy to encipher or decipher a message

2. It must be computationally infeasible to derive the private key from the public key

3. It must be computationally infeasible to determine the private key from a chosen plaintext attack


Public-Key Cryptography public-key/two-key/asymmetric

cryptography involves the use of two keys: a public-key, which may be known by

anybody, and can be used to encrypt messages, and verify signatures

a private-key, known only to the recipient, used to decrypt messages, and sign (create) signatures


Public-Key Cryptography is asymmetric because

those who encrypt messages or verify signatures cannot decrypt messages or create signatures


Public-Key Cryptography


Why Public-Key Cryptography? developed to address two key issues:

key distribution – how to have secure communications in general without having to trust a KDC with your key

digital signatures – how to verify a message comes intact from the claimed sender

public invention due to Whitfield Diffie & Martin Hellman at Stanford in 1976 known earlier in classified community


Public-Key Applications can classify uses into 3 categories:

key exchange (of session keys) encryption/decryption (provide

secrecy) digital signatures (provide

authentication) some algorithms are suitable for all

uses, others are specific to one


Security of Public Key Schemes like private key schemes brute force

exhaustive search attack is always theoretically possible

but keys used are too large (>512 bits) not comparable to symmetric key sizes

security relies on a large enough difference in difficulty between easy (en/decrypt) and hard (to cryptanalyze) problems

requires the use of very large numbers hence is slow compared to secret key

schemes


Diffie-Hellman Compute a common, shared key

Called a symmetric key exchange protocol

Based on discrete logarithm problem Given integers n and g and prime

number p, compute k such that n = gk mod p

Solutions known for small p Solutions computationally infeasible as

p grows large – hence, choose large p


Diffie-Hellman Key Exchange a public-key distribution scheme

cannot be used to exchange arbitrary messages

rather it can establish a common key known only to the two participants

value of key depends on the participants (and their private and public key information)

based on exponentiation in a finite (Galois) field (modulo a prime or a polynomial) - easy

security relies on the difficulty of computing discrete logarithms (similar to factoring) – hard


Diffie-Hellman SetupGenerating the Diffie-Hellman public

key The Diffie-Hellman system allows Alice

and Bob to agree on a key even when Eve is listening to everything they say to each other.

Alice and Bob need to agree on a prime number p, which they can do by simply sending it to each other.

Eve is allowed to learn this number p. In practice the number p is often simply advertised somewhere public.


Generators & Public Key Given a prime number p, it is possible to come up with a

number g (the so-called generator) with a very interesting property. Every number between 1 and p-1 can be written as a power of g when calculating modulo p. For example, using p = 5 the generator is 2, because

20 = 1 21 = 2 22 = 4 23 = 3 (because 8 = 3 mod 5)

Alice and Bob agree in the same way on a generator g for the numbers between 1 and p-1.

The numbers p and g serve as the public key.


Diffie-Hellman Key Exchange

Shared key, public communication No authentication of partners What’s involved?

P is a prime (about 512 bits), and g < p

P and g are publicly known


Diffie-Hellman Key Exchange

ProcedureAlice Bobpick secret Sa randomly pick secret Sb

randomlycompute TA=gSa mod p compute

TB=gSb mod p

send TA to Bob send TB to Alice

compute TBSa mod p compute TA

Sb mod p


Alice and Bob reached the same secret gSaSb mod p, which is then used as the shared key.


DH Security - Discrete Logarithm Is Hard

T = gs mod p Conjecture: given T, g, p, it is

extremely hard to compute the value of s (discrete logarithm)


Diffie-Hellman Scheme

Security factors Discrete logarithm very difficult. Shared key (the secret) itself never

transmitted.


Disadvantages: Expensive exponential operation

DoS possible. The scheme itself cannot be used to

encrypt anything – it is for secret key establishment.

No authentication, so you can not sign anything …


Bucket Brigade Attack...Man In The Middle

Alice Trudy BobgSa=123 gSx =654 gSb

=255 123 --> 654 -->

<--654 <--255 654Sa=123Sx 255Sx=654Sb

Trudy plays Bob to Alice and Alice to Bob


Encryption With Diffie-Hellman

Everyone computes and publishes <p, g, T> T=gS mod p

Alice communicates with Bob: Alice

Picks a random secret Sa

Computes gbSa mod pb

Use Kab = TbSa mod pb to encrypt message

Send encrypted message along with gbSa

mod pb


Diffie-Hellman Bob

(gbSa)Sb mod pb = (gb

Sb)Sa mod pb = TbSa

mod pb = Kab

Use Kab to decrypt


Example of an attack

N=11 g=7 Alice chooses x=3 Bob chooses y=9 Trudy chooses x=8 and y=6


Alice computes 73 mod 11 =2 Bob computes 79 mod 11 =8 Trudy computes 78 mod 11 =9 and 76 mod 11 = 4 Alice sends 2 to Bob….Trudy

intercepts and sends 9 Bob sends 8 to Alice….Trudy

intercepts and sends 4


Alice computes K1= 43 mod 11 =9 Bob computes K2= 99 mod 11 =5

Trudy computes K1 = 88 mod 11 =5 And K2= 26 mod 11 = 9

Thus Trudy is man-in-the-middle……..

RAITM. Chatterjee1 Public Key Algorithms ……... RAITM. Chatterjee2 Public Key Cryptography Two keys Private key known only to individual Public key available.

Documents

common key

public key information

shared key

key issues

appropriate key

diffiehellman public

keys private key

key digital signatures